Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the polynomials. \((4 x-6 y)(6 x-4 y)\)

Step-by-Step Solution

Answer

The product of \((4x-6y)(6x-4y)\) is \(24x^2 - 52xy + 24y^2\).

Step 1: Apply the Distributive Property

To multiply the polynomials, apply the distributive property also known as the FOIL method (First, Outside, Inside, Last): Multiply each term in the first polynomial by each term in the second polynomial. Start by multiplying the first terms of each polynomial: \(4x \times 6x = 24x^2\).

Step 2: Multiply the Outside Terms

Next, multiply the outside terms of the two binomials: \(4x \times -4y = -16xy\).

Step 3: Multiply the Inside Terms

Multiply the inside terms: \(-6y \times 6x = -36xy\).

Step 4: Multiply the Last Terms

Finally, multiply the last terms of each binomial: \(-6y \times -4y = 24y^2\).

Step 5: Combine Like Terms

Now, combine the intermediate results from steps 1-4. Combine the \(-16xy\) and \(-36xy\), which are like terms: \(24x^2 - 16xy - 36xy + 24y^2 = 24x^2 - 52xy + 24y^2\).

Key Concepts

Distributive Property

The distributive property is a key concept in algebra, allowing us to multiply a single term across terms inside a parentheses. In mathematical operations, it provides us a way to simplify expressions. Here's how it works:

• Multiply each term inside the first set of parentheses by each term inside the second set.
• This property helps us expand expressions like \((a+b)(c+d)\), breaking it down to make calculations easier.

This process turns complex polynomial multiplication into a step-by-step procedure. By distributing each element, you're ensuring that each component is correctly accounted for, keeping the calculations organized. In the example provided, we handle each term of the first polynomial with each term of the second polynomial.

FOIL Method

The FOIL method is a handy acronym to remember when multiplying two binomials. FOIL stands for First, Outside, Inside, and Last. This method is particularly useful when dealing with expressions of the form \((a + b)(c + d)\):

• First: Multiply the first terms from each binomial.
• Outside: Multiply the outer terms in the expression.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

For example, when multiplying \((4x - 6y)(6x - 4y)\):
First: \(4x imes 6x = 24x^2\)
Outside: \(4x imes -4y = -16xy\)
Inside: \(-6y imes 6x = -36xy\)
Last: \(-6y imes -4y = 24y^2\)
This systematic approach simplifies the process and ensures that no terms are left out.

Combining Like Terms

After using the FOIL method to expand, the next crucial step is combining like terms. Like terms share the same variable part and can be added or subtracted from one another.
This step helps simplify the expression into its simplest form.

• For instance, the expression \(24x^2 - 16xy - 36xy + 24y^2\) features like terms \(-16xy\) and \(-36xy\).
• Combine these to get: \(24x^2 - 52xy + 24y^2\).

Combining like terms reduces clutter and provides a more concise expression, essential for clarity in both academic and real-world applications.

Binomials

A binomial is a polynomial that consists of exactly two terms. They serve as a basic building block in algebra and are often the focus when learning about algebraic multiplication.

• Binomials can take the form of expressions like \((a+b)\) or \((m-n)\).
• When multiplying binomials, tools like the FOIL method and distributive property become invaluable.

Understanding binomials is essential because they frequently appear in algebraic problems and require specific strategies for solving. The example \((4x - 6y)(6x - 4y)\) we discussed involves multiplying binomials, a common task in expanding polynomial expressions. Mastery of binomials simplifies more complex polynomial operations, establishing a solid foundation for further mathematical explorations.